Highly Inclined and Eccentric Massive Planets I

A&A 555, A124 (2013) Astronomy DOI: 10.1051/0004-6361/201220310 & c ESO 2013 Astrophysics

Highly inclined and eccentric massive planets I. Planet-disc interactions B. Bitsch1, A. Crida1, A.-S. Libert2,3, and E. Lega1

1 University of Nice-Sophia Antipolis / CNRS / Observatoire de la Côte d’Azur, Laboratoire Lagrange UMR 7293, Boulevard de l’Observatoire, BP4229, 06304 Nice Cedex 4, France e-mail: [email protected] 2 NaXys, Department of Mathematics, University of Namur, 8 rempart de la Vierge, 5000 Namur, Belgium 3 Observatoire de Lille (LAL-IMCCE), CNRS-UMR 8028, 1 impasse de l’Observatoire, 59000 Lille, France Received 30 August 2012 / Accepted 11 May 2013 ABSTRACT

Context. In the solar system, planets have a small inclination with respect to the equatorial plane of the Sun, but there is evidence that in extrasolar systems the inclination can be very high. This spin-orbit misalignment is unexpected, as planets form in a protoplanetary disc supposedly aligned with the stellar spin. It has been proposed that planet-planet interactions can lead to mutual inclinations during migration in the protoplanetary disc. However, the effect of the gas disc on inclined giant planets is still unknown. Aims. In this paper we investigate planet-disc interactions for planets above 1 MJup. We check the influence of three parameters: the inclination i, eccentricity e,andmassMp of the planet. This analysis also aims at providing a general expression of the eccentricity and inclination damping exerted on the planet by the disc. Methods. We perform three-dimensional numerical simulations of protoplanetary discs with embedded high-mass planets on fixed orbits. We use the explicit/implicit hydrodynamical code NIRVANA in 3D with an isothermal equation of state. Results. We provide damping formulae for i and e as a function of i, e,andMp that fit the numerical data. For highly inclined massive −4 planets, the gap opening is reduced, and the damping of i occurs on time-scales of the order of 10 deg/year · Mdisc/(0.01 M) with the damping of e on a smaller time-scale. While the inclination of low planetary masses (<5 MJup) is always damped, large planetary masses with large i can undergo a Kozai-cycle with the disc. These Kozai-cycles are damped through the disc in time. Eccentricity is generally damped, except for very massive planets (Mp ∼ 5 MJup) where eccentricity can increase for low inclinations. So the dynamics tends to a final state: planets end up in midplane and can then, over time, increase their eccentricity as a result of interactions with the disc. Conclusions. The interactions with the disc lead to damping of i and e after a scattering event of high-mass planets. If i is sufficiently reduced, the eccentricity can be pumped up because of interactions with the disc. If the planet is scattered to high inclination, it can undergo a Kozai-cycle with the disc that makes it hard to predict the exact movement of the planet and its orbital parameters at the dispersal of the disc. Key words. accretion, accretion disks – planets and satellites: formation – hydrodynamics – planet-disk interactions

1. Introduction Chatterjee et al. 2008; Juric´ & Tremaine 2008). These works assume that unstable crowded systems are formed, and undergo In the solar system, the orbits of all the planets are nearly copla- planet-planet scattering after a relatively short time when the nar (within 4 degrees, except for Mercury). The ecliptic (the gas nebula dissipates. However, recent work suggests that un- plane of the Earth’s orbit) is also close to the equatorial plane = ◦ stable systems reach instability while still embedded in the gas of the Sun: the spin-orbit misalignment is only β⊕ 7.5 .The disc (Lega et al. 2013). A second process is planet-planet inter- low inclination of the massive planets with respect to the ecliptic actions during migration in the protoplanetary disc (Thommes is normally taken as an indication that planets form within a flat- & Lissauer 2003; Libert & Tsiganis 2009, 2011a,b). During the tened protoplanetary disc, itself closely aligned with the stellar gas-driven migration, the system can enter an inclination-type equator. The newly discovered Kepler-30 system (Sanchis-Ojeda resonance or the resonant configuration becomes unstable as the et al. 2012) is even flatter, and confirms this view. However, resonance excites the eccentricities of the planets and planet- exo-planets with strong spin-orbit misalignment have been de- ◦ planet scattering sets in. All this affirms the need for a bet- tected (e.g. β>50 ; Moutou et al. 2011a,b; Hébrard et al. 2011; ter understanding of the interactions between giant planets and Simpson et al. 2011). Considering that the plane of the past pro- a gaseous protoplanetary disc when the orbit of the former is toplanetary disc should be identical to the present stellar equa- 1 highly inclined with respect to the midplane of the later. Here, tor , the orbital plane of these planets must have been changed we study this phenomenon, in detail. by some mechanism. One process generally invoked to explain inclined orbits is Tanaka & Ward (2004) have shown in linear studies that the scattering by multiple planets in the system after the protoplan- inclination of a low-mass planet embedded in a disc is exponen- etary disc has dissipated (e.g. Marzari & Weidenschilling 2002; tially damped by planet-disc interactions for any non-vanishing inclination. Such results are formally valid only for i H/r. 1 This is generally accepted, but is actually the subject of debate (see However, numerical simulations of more highly inclined plan- e.g. Cébron et al. 2011; Batygin 2012). ets have shown that the exponential damping might be valid up Article published by EDP Sciences A124, page 1 of 13 A&A 555, A124 (2013) to i ≈ 2H/r. If the planet has an even greater inclination, the highly inclined planets (i = 75.0◦). Our test simulations, how- damping rates deviate from being exponential and it can be fit- ever, show that this algorithm can also be used in highly inclined tedbyadi/dt ∝ i−2 function (Cresswell et al. 2007; Bitsch & planets, see Appendix. A.Herewetreatthediscasaviscous Kley 2011). However, for high-mass planets, the linear regime medium in the locally isothermal regime. We do not use radia- is no longer valid. Marzari & Nelson (2009) considered Jovian- tion transport, as we focus here on high-mass planets that open type planets on inclined and eccentric orbits. They find highly agapinsideadisc,wheretheeffects of heating and cooling of inclined and eccentric planets with Jovian masses lose their in- the disc are much less important than for low-mass planets (Kley clination and eccentricity very quickly (on a time-scale of the et al. 2009). A more detailed description of the used code can be order of 103 years) when entering the disc again (when i < H/r). found in Kley et al. (2009). Since a highly inclined planet is only slightly disturbed by the accretion disc (and vice versa), this kind of planet is only able to open a gap in the disc when the inclination drops to i < 10.0◦. 2.1. Smoothing of the planetary potential Planet-disc interactions also influence the eccentricity of em- An important issue in modelling planetary dynamics in discs is bedded planets, as has been shown by Goldreich & Tremaine the gravitational potential of the planet since this has to be ar- (1980). It has been suggested, by performing linear analysis, that tificially smoothed to avoid singularities. While in two dimen- the planetary eccentricity can be increased through planet-disc sions a potential smoothing takes care of the otherwise neglected interaction under some conditions (Goldreich & Sari 2003; Sari vertical extension of the disc, in 3D simulations the most accu- & Goldreich 2004; Moorhead & Adams 2008). They estimate rate potential should be used. As the planetary radius is much that eccentric Lindblad resonances can cause eccentricity growth smaller than a typical grid cell, and the planet is treated as a for gap-forming planets. However, numerical simulations show point mass, a smoothing of the potential is required to ensure that eccentricity in the disc is damped for a variety of masses numerical stability. (Cresswell et al. 2007; Moorhead & Ford 2009; Bitsch & Kley In Kley et al. (2009)twodifferent kinds of planetary poten- 2010). tials for 3D discs have been discussed. The first is the classic For very-high-mass planets, on the other hand an eccentric sm-potential instability in the disc can arise (Kley & Dirksen 2006). In turn, GMp Φsm = − · this eccentric disc can possibly increase the planetary eccentric- p (1) 2 2 ity (Papaloizou et al. 2001; D’Angelo et al. 2006). However, d + sm this process can only explain the eccentricity of very massive Here M is the planetary mass, and d = |r − r | denotes the dis- ≈ P P ( 5–10 MJup) planets. Xiang-Gruess & Papaloizou (2013)have tance of the disc element to the planet. This potential has the recently studied the interactions between Jupiter-mass planets advantage that it leads to very stable evolutions when the param- and circumstellar discs as well. However, they did not consider eter sm is a significant fraction of the Roche radius. The disad- planets on eccentric orbits and they were using SPH simulations, vantage is that for smaller sm, which would yield a higher accu- while we use a grid-based code. racy at larger d, the potential becomes very deep at the planetary In this paper, we investigate planet-disc interactions for plan- position. Additionally, the potential differs from the exact 1/r ff ets above 1 MJup, considering di erent inclination and eccentric- potential even for medium to larger distances d from the planet. ity values. Our analysis also aims at deriving a formula for the To resolve these problems at small and large d simultane- change of eccentricity and inclination due to planet-disc interac- ously, the following cubic-potential has been suggested (Klahr tions, in order to study the long-term evolution of systems with & Kley 2006; Kley et al. 2009) massive planets. Indeed, long-term evolution studies of plane- ⎧ ⎪ 4 3 ⎪ GMp d d d tary systems cannot be done with hydrodynamical simulations, ⎨ − − 2 + 2 for d ≤ rsm . Φcub = ⎪ d rsm rsm rsm as the computation time is too long, and N-Body codes that con- p ⎪ (2) ⎩ − GMp d > r . sider the gravitational effects only are used. A correct damping d for sm rate of eccentricity and inclination is needed in order to simu- The construction of the planetary potential is such that for dis- late the evolution correctly. This study will be the topic of our tances larger than rsm the potential matches the correct 1/r po- Paper II. tential. Inside this radius (d < rsm) it is smoothed by a cubic We use isothermal three-dimensional (3D) simulations to de- polynomial. This potential has the advantage of exactness out- termine the change of inclination and eccentricity due to planet side the specified distance rsm, while being finite inside. disc interactions. In Sect. 2 we describe the numerical methods For 1 MJup and 5 MJup we use the cubic potential with rsm = used, as well as the procedure to calculate the forces acting on 0.8RH.Forthe10MJup planet, we use the sm-potential with the embedded planets to determine di/dt and de/dt. In Sect. 3 we rsm = 0.8RH, with the Hill radius RH given by show di/dt and de/dt as a function of inclination i and eccentric- 1/3 Mp ity e, and provide fitting formulae. Additionally an observed os- RH = ap , (3) cillatory behaviour is discussed in this section. The implications 3M for single-planet systems are shown in Sect. 4. where ap is the semi major axis of the planet, and M is the mass of the central star. 2. Physical modelling As the planetary mass increases, so does the amount of material accumulated near the planet. In order to resolve the gra- The protoplanetary disc is modelled as a 3D, non-self- dients of density in that region correctly, a much higher resolu- gravitating gas whose motion is described by the Navier-Stokes tion is required. Therefore, we change the cubic potential to the equations. We use the code Nirvana (Ziegler & Yorke 1997; sm-potential for the 10 MJup planet. For the torque acting on the Kley et al. 2001), which uses the FARGO-algorithm (Masset planets, the consequences are minimal, as we use a torque cut- 2000) and was described in our earlier work on planets on in- off function in the Hill sphere of the planet, as described below. clined orbits (Bitsch & Kley 2011). We note that the use of Additional information regarding the smoothing length can be the FARGO-algorithm may not be straight forward in the case of found in Appendix A.

A124, page 2 of 13 B. Bitsch et al.: Highly inclined and eccentric massive planets. I.

2.2. Initial setup to disregard material that is possibly gravitationally bound to the planet (Crida et al. 2009). Here we assume b = 0.8, as a change The 3D (r,θ,φ) computational domain consists of a complete in b did not influence the results significantly (Kley et al. 2009). annulus of the protoplanetary disc centred on the star, extend- = = = = If a small disturbing force dF (given per unit mass) due to ing from rmin 0.2tormax 4.2 in units of r0 aJup 5.2 the disc is acting on the planet, the planet changes its orbit. This AU, where we put the planet. The planet is held on a fixed or- small disturbing force dF may change the planetary orbit in size bit during the evolution. The eccentricity of the planet can be (semi-major axis a), eccentricity e, and inclination i. The incli- e0 = 0.0, e0 = 0.2, or e0 = 0.4. We use 390 × 48 × 576 active × × nation i gives the angle between the orbital plane and an arbitrary cells for the simulations with 1 MJup and 260 32 384 active = ◦ ffi fixed plane, which is in our case the equatorial plane (θ 90 ), cells for 5 MJup and 10 MJup. This resolution is su cient, as we which corresponds to the midplane of the disc. Only forces ly- still resolve the horseshoe width with a few grid cells for all plan- ing in the orbit plane can change the orbit’s size and shape, while etary masses.√ The horseshoe width is defined for large planets as these forces cannot change the orientation of the orbital plane. In = 1/3 xs 12aP(q/3) (Masset et al. 2006), where q is the planet- Burns (1976) the specific disturbing force is written as star mass ratio. Tests regarding the numerical resolution can be found in Appendix A. dF = R + T + N = ReR + TeT + NeN, (6) In the vertical direction, the annulus extends 7◦ below and above the disc’s midplane, meaning 83◦ <θ<97◦.Hereθ where each e represents the relevant orthogonal component of denotes the polar angle of our spherical polar coordinate sys- the unit vector. The perturbing force can be split into these com- tem measured from the polar axis, therefore the midplane of the ponents: R is radially outwards along r; T is transverse to the disc is at θ = 90.0◦. We use closed boundary conditions in the radial vector in the orbit plane (positive in the direction of motion of the planet); and N is normal to the orbit planet in the radial and vertical directions. In the azimuthal direction, peri- × odic boundary conditions are used. The central star has one so- direction R T. Burns (1976) finds for the change of inclination lar mass M∗ = M , and the total disc mass inside [rmin, rmax] = = is Mdisc 0.01M . The aspect ratio of the disc is H/r 0.05. di aN cos ξ = 2 Ω = We use an α prescription of the viscosity, where ν αcs / K , (7) = Ω dt H (Shakura & Sunyaev 1973) with α 0.005 ; K is the Kepler where the numerator is the component of the torque which ro- frequency; cs = P/ρ denotes the isothermal sound speed, tates the specific angular momentum H = r × ˙r about the line of P the pressure, ρ the volume density of the gas, and H = cs/Ω. The models are initialised with constant temperatures on nodes (and which thereby changes the inclination of the orbital cylinders with a profile T(s) ∝ s−1 with s = r sin θ. This yields a plane). The specific angular momentum H is defined as constant ratio of the disc’s vertical height H to the radius s.The 2 initial vertical density stratification is given approximately by a H = GMap(1 − e ). (8) Gaussian The angle ξ is related to the true anomaly f by f = ξ − ω, − 2 2 (π/2 θ) r with ω being the argument of periapsis and ξ describes the angle ρ(r,θ) = ρ0(r)exp − · (4) 2H2 between the line of nodes and the planet on its orbit around the star. For the case of circular orbits, the argument of periapsis ω ∝ −1.5 Here, the density in the midplane is ρ0(r) r which leads is zero. Σ ∝ −1/2 to a (r) r profile of the vertically integrated surface den- The change of eccentricity is given by Burns (1976)as sity. In the radial and θ-direction we set the initial velocities to 1/2 zero, while for the azimuthal component the initial velocity uφ is de a(1 − e2) given by the equilibrium of gravity, centrifugal acceleration and = R sin f + T(cos f + cos ) , (9) dt GM the radial pressure gradient. This corresponds to the equilibrium configuration for a purely isothermal disc with constant viscos- where is the eccentric anomaly, which is given by ity. However, as the massive planets in the disc start to open gaps, e + cos f the density and surface density profile get distorted. cos = · (10) 1 + e cos f 2.3. Calculation of forces With this set of equations, we are able to calculate the forces acting on planets on fixed orbits and determine di/dt and de/dt. To determine the change of orbital elements for planets on fixed inclined orbits, we follow Burns (1976) and compute the forces as described in Bitsch & Kley (2011). The gravitational torques 3. Planets on inclined and eccentric orbits and forces acting on the planet are calculated by integrating over the whole disc, where we apply a tapering function to exclude In this section we investigate the changes of the planetary orbit the inner parts of the Hill sphere of the planet. Specifically, we due to planet-disc interactions. The planets are put in fixed orbits = ◦ = ◦ use the smooth (Fermi-type) function with inclinations ranging from i0 1.0 to i0 75 , with a total ff of ten di erent inclinations. For each inclination we also adopt −1 d/R − b three different eccentricities, which are e0 = 0.0, e0 = 0.2and f (d) = exp − H + 1 (5) = b b/10 e0 0.4. We note that the orbit of highly inclined planets is not em- which increases from 0 at the planet location (d = 0) to 1 out- bedded completely in the hydrodynamical grid, since the grid is ◦ side d ≥ RH with a midpoint fb = 1/2atd = bRH,i.e.the only extended up to 7 above and below midplane. However, the quantity b denotes the torque-cutoff radius in units of the Hill density distribution in the vertical direction follows a Gaussian radius. This torque-cutoff is necessary to avoid large, probably profile and for an aspect ratio of 0.05 we are at about 2.5σ at 7◦ noisy contributions from the inner parts of the Roche lobe and so that the contribution of the gas can be neglected at larger θ.

A124, page 3 of 13 A&A 555, A124 (2013)

10000 -9 i0 = 1.0 i = 20.0 0.4 0 -10 i0 = 75.0

i0 = 1.0, e0=0.4 ) 1000 0.2 3

initial ] -11 Jup 0 -12 in g/cm 2 100 ρ z in [a -0.2 -13 log ( in g/cm

Σ 10 -14 -0.4 -15 1 0.5 1 1.5 2 2.5 3 3.5 4 r [aJup] -9 0.1 0.4 0.5 1 1.5 2 2.5 3 3.5 4 -10

r [a ] ) Jup 0.2 3 ] -11 Fig. 1. Surface density for disc simulations with 10 MJup planets in cir- Jup 0 cular and eccentric orbits with diﬀerent inclinations. The surface den- in g/cm -12 ρ sity is plotted after 400 planetary orbits. The evolution has reached an z in [a -0.2 equilibrium state, meaning that the surface density does not change in log ( time any more. -13 -0.4 -14 0.5 1 1.5 2 2.5 3 3.5 4 3.1. Gaps in discs r [aJup] The criterion for gap opening depends on the viscosity, the pres- -9 sure, and the planetary mass (Crida et al. 2006). Giant planets 0.4 -9.5

(M 0.5 M ) are generally massive enough to split the disc. -10 ) Jup 0.2 3 However, the inclination of a giant planet plays a very important ] -10.5 Jup role in opening a gap as well, as can be seen in Fig. 1,where 0 -11 in g/cm we display the surface density profile of discs with embedded ρ z in [a -11.5 10 MJup planets on different inclinations. -0.2 -12 log ( Clearly, a lower inclination produces a much wider and deeper gap inside the disc. For larger inclinations, the gap open- -0.4 -12.5 ing is reduced, as the planet spends less and less time inside the -13 0.5 1 1.5 2 2.5 3 3.5 4 disc to push material away from its orbit. Additionally, eccentric r [a ] planets open up gaps that are less deep than their circular counter Jup parts. This effect is very important for the damping of inclina- Fig. 2. Density (in g/cm3)ofar − θ-slice through the disc at the az- tion and eccentricity, as an open gap inside the disc prolongs the imuth of an embedded 10 MJup planet on a fixed circular inclined orbit = ◦ = ◦ = ◦ damping time-scale of inclination (Bitsch & Kley 2011) and of with i0 1.0 (top), i0 20 (middle), and i0 75.0 (bottom). The eccentricity (Bitsch & Kley 2010). Gap opening also indicates planet is at its lowest point in orbit (lower culmination) at the time of the snapshot, which was taken after 400 planetary orbits. We note the that linear analysis of the situation is no longer applicable. ff In Fig. 2 we present slices in the x − z-plane for the disc’s slightly di erent colour scale for each plot. The black line indicates the ◦ ◦ midplane of the grid to which the inclination of the disc is measured density for 10 MJup planets on inclinations of 1 ,20,and (see Sect. 3.2). 75◦ degrees. The inclinations correspond to those shown in the surface density plot (Fig. 1). Clearly the depth of the gap shown in the surface density is reflected in the 2D plots. Additionally, the density structures show no effects at the upper and lower For low planetary inclinations, the influence of the planet boundaries because of boundary conditions, indicating that an ◦ on the eccentricity of the protoplanetary disc is greater than on opening angle of 7 is sufficient for our simulations. high planetary inclinations, simply, because the planet is closer to midplane and can therefore influence the eccentricity of the 3.2. Change of the disc structure disc more strongly by pushing the material away. The eccentricity increase of the disc is stronger for planets in circular orbits It has been known since several years that massive planets are than for planets that are already in an eccentric orbit. For highly able not only to open up a gap in the disc, but are also able inclined planets, the situation is reversed. The disc is most eccen- to change the shape of the whole disc by turning it eccentric tric for planets that are already in an eccentric orbit and the disc (Papaloizou et al. 2001; Kley & Dirksen 2006). Additionally, is less eccentric for planets in circular orbits. Additionally, the the inclination of the disc will change due to the interactions eccentricity of the disc is highest close to the planet and drops with the inclined planet. In this section, we discuss the impact of with distance from the planet, independent of the inclination of a massive planet on the eccentricity and inclination of the disc. the planet. ◦ In Fig. 3 we display the eccentricity (top) and inclination The inclination of the disc for the i0 = 1 planets is greater (bottom) of the disc interacting with a 10 MJup planet with differ- mostly around the planet’s location (at r = 1.0aJup) because the ent inclinations (1◦ and 75◦) and eccentricities. The calculations influence of the planet is strongest there. The inclination of the for deriving the eccentricity and inclination of the disc can be disc can be larger than the inclination of the planet. This is pos- found in Appendix B. sible because the planet opens a gap inside the disc and pushes

A124, page 4 of 13 B. Bitsch et al.: Highly inclined and eccentric massive planets. I.

0.3 0.0005 i0 = 1.0, e0=0.0 i0 = 1.0, e0=0.2 0 i = 1.0, e =0.4 0.25 0 0 i0 = 75.0, e0=0.0 -0.0005 i0 = 75.0, e0=0.2 i0 = 75.0, e0=0.4 0.2 -0.001

] -0.0015 s it b

disc 0.15 -0.002 e -0.0025 0.1 de/dt [1/or -0.003

0.05 -0.0035 -0.004 1 MJup, e=0.2 1 MJup, e=0.4 0 -0.0045 ﬁt 1 M , e=0.2 0.5 1 1.5 2 2.5 3 3.5 4 Jup ﬁt 1 MJup, e=0.4 0.0004 r [aJup] 0.0002 2.5 i0 = 1.0, e0=0.0 0 i0 = 1.0, e0=0.2 i0 = 1.0, e0=0.4 -0.0002 2 i0 = 75.0, e0=0.0 i0 = 75.0, e0=0.2 -0.0004

i0 = 75.0, e0=0.4 ] s

it -0.0006 1.5 b -0.0008

in deg -0.001 de/dt [1/or disc

i 1 -0.0012

-0.0014 0.5 -0.0016 5 MJup, e=0.2 5 MJup, e=0.4 -0.0018 fit 5 MJup, e=0.2 0 fit 5 MJup, e=0.4 0.5 1 1.5 2 2.5 3 3.5 4 0.0006 10 MJup, e=0.2 10 M , e=0.4 r [aJup] 0.0004 Jup fit 10 MJup, e=0.2 0.0002 fit 10 M , e=0.4 Fig. 3. Eccentricity (top) and inclination (bottom) of the disc with a Jup 0 10 MJup planet influencing the disc structure after 400 planetary orbits. -0.0002 ] s it

b -0.0004 the material away from the planet (Fig. 1, top), which can also -0.0006 be seen in the 2D density configuration (top of Fig. 2). In the -0.0008 de/dt [1/or outer parts of the disc the disc remains non-inclined. -0.001 For planets with high inclinations the situation is slightly dif- -0.0012 ferent than for planets with low inclinations. The maximum of -0.0014 inclination is lower and there is no distinct maximum of inclina- -0.0016 tion visible inside the planetary orbit (r < 1.0aJup) compared to the case of low inclinations. However, the outer parts of the disc 0 10 20 30 40 50 60 70 80 show a non-zero inclination (which has a tendency to be larger inclination in degrees for larger planetary eccentricities), which was not visible for the low-inclination planets. Additionally, they show a small peak of Fig. 4. Change of eccentricity de/dt for planets with 1 MJup,5MJup,and 10 MJup with different eccentricities. Points are results from numerical inclination at r ≈ 1.25aJup. simulations, while lines indicate the fitting of the data. The 1 MJup planets have been evolved for 200 planetary orbits, the 5 MJup and 10 MJup 3.3. Change of orbital parameters planets have been evolved for 400 orbits. The forces used to calculate the data points have been averaged over 40 planetary orbits for all 3.3.1. Eccentricity simulations. As stated in Sect. 2.3, the forces acting on a planet on a fixed orbit can be calculated and then used to determine a rate of in the past for coplanar low-mass planets (Cresswell et al. 2007; change for the inclination and eccentricity. The damping rates Bitsch & Kley 2010) and for high-mass planets (Papaloizou et al. are taken when the planet-disc interactions are in an equilibrium 2001; Kley & Dirksen 2006). ◦ state and do not change on average any more. The damping given For low inclinations (i0 < 10 ) the damping of eccentric- by Eq. (7) varies strongly within the time of an orbit and slightly ity is stronger than for larger inclinations in the case of 1 MJup. from one orbit to an other. Thus, we averaged it over 40 plane- The maximum damping rate is also dependent on the initial ec- tary orbits. centricity e0, where a larger e0 provides a faster damping. The In Fig. 4 we present the change of eccentricity de/dt for plan- damping of eccentricity is reduced significantly for larger incli- ◦ ets of 1 MJup,5MJup, and 10 MJup on orbits with different inclinations i0 > 20 . As soon as the planet is no longer embedded nations and eccentricities. The change of de/dt has been studied in the disc, the damping reduces, as it is most efficient when the

A124, page 5 of 13 A&A 555, A124 (2013) planet is inside the disc and not high above or below the disc for 0 most of its orbit. ◦ -0.02 For low inclinations (i0 < 10 ) and low eccentricities (e0 < 0.2), the 5 MJup planet opens up a large gap inside the disc. As -0.04 the planet opens a gap inside the disc the damping is reduced be- ] cause there is less material close to the planet to damp its orbit. s -0.06 it For large initial eccentricities (e0 = 0.4), an increase of eccen- b tricity is observable for low planetary inclinations. But for higher -0.08 inclinations, the damping of eccentricity increases as well, be- -0.1 cause the planet does not open up such a deep gap (Fig. 1). di/dt [deg/or ◦ However, for i0 > 40 the damping of eccentricity becomes smaller again, because the planet spends less and less time in the -0.12 1 MJup, e=0.0 1 MJup, e=0.2 midplane of the disc where most of the disc material is, which is 1 MJup, e=0.4 -0.14 fit 1 MJup, e=0.0 responsible for damping. fit 1 MJup, e=0.2 fit 1 MJup, e=0.4 For even higher masses (10 MJup), we observe an eccentric- 0.01 ity increase for low planetary inclinations for all non-zero eccen- ◦ tricities. But for larger inclinations (i0 > 15 ), the eccentricity is ◦ ◦ 0 damped again. The largest value of damping is at i0 ≈ 30 −50 , depending on the planet’s eccentricity and is then reduced for higher inclinations again, following the trend described for the -0.01 ]

5 MJup planet. s it For large planets with low inclinations, the eccentricity of b -0.02 the planet can rise, which has been observed in Papaloizou et al. (2001)andKley & Dirksen (2006). Papaloizou et al. (2001) -0.03

stated that if the planet opens up a large gap, the m = 2spi- di/dt [deg/or ral wave at the 1:3 outer eccentric Lindblad resonance becomes -0.04 dominant (because the order 1 resonances lie inside the gap) and 5 MJup, e=0.0 induces eccentricity growth. However, they found an eccentricity 5 MJup, e=0.2 -0.05 5 MJup, e=0.4 fit 5 M , e=0.0 increase only for MP > 20 MJup, while our simulations indicate Jup ff fit 5 MJup, e=0.2 it clearly already for MP > 5 MJup (Fig. 4, bottom). The di er- fit 5 MJup, e=0.4 ences between their 2D simulations and our 3D simulations can 0.005 be the cause of the change in the required planetary mass for 0 eccentricity growth. Additionally, by embedding a high-mass planet inside a disc, -0.005 the disc can become eccentric, as shown in Sect. 3.2. The disc’s

] -0.01 s eccentricity is dependent on the planet’s inclination and slightly it b dependent on its eccentricity as well (see Fig. 3). -0.015 It seems that the coupling between a large disc eccentricity ◦ -0.02 at r ≈ 1−1.5aP and a large planetary eccentricity (the i0 = 1 with e = 0.4 case) results in a large force on the planet. This ef- di/dt [deg/or -0.025 fect is increased as the planet in an eccentric orbit opens a small 10 MJup, e=0.0 -0.03 10 MJup, e=0.2 gap leaving more material at that location. This leads then to a 10 MJup, e=0.4 greater increase of eccentricity for highly eccentric planets, com- -0.035 fit 10 MJup, e=0.0 fit 10 MJup, e=0.2 pared to those with small eccentricity. fit 10 MJup, e=0.4 0 10 20 30 40 50 60 70 80 3.3.2. Inclination inclination in degrees In Fig. 5 we present the rate of change of inclination di/dt,pre- Fig. 5. Change of inclination di/dt for planets with 1 MJup,5MJup,and 10 M with different eccentricities. di/dt is in degrees per orbit at the sented in degrees per orbit, for planets with different masses and Jup planet’s location rP = 1.0aJup. Points are results from numerical simula- different eccentricities. For 1 MJup the inclination is damped for ◦ tions, while lines indicate the fitting of the data. The 1 MJup planets have all initial inclinations. For increasing inclinations with i0 < 15 been evolved for 200 planetary orbits, the 5 MJup and 10 MJup planets (smaller for increasing eccentricity), the damping of inclination have been evolved for 400 orbits. The forces used to calculate the data increases. This increase is nearly linear, as has been shown for points have been averaged over 40 planetary orbits for all simulations. low-mass planets in theory (Tanaka & Ward 2004) and in numerical simulations (Cresswell et al. 2007; Bitsch & Kley 2011). The rates of inclination damping for zero-eccentricity planets are comparable to those stated in Xiang-Gruess & Papaloizou there is a significant difference for high inclined and low ec- ◦ (2013). centric planets: the inclination is not damped if i0 > 50 ,but ◦ For i0 > 15 , the damping rate of the inclination is a decreas- it increases for e0 < 0.1. This behaviour will be discussed in ing function of inclination ; this is consistent with the planet-disc Sect. 3.4. interaction being weaker when the planet spends more time far- The 10 MJup planet shows the same general behaviour as the ther from the midplane. 5 MJup planet, but the inclination increase already sets in at i0 ◦ For 5 MJup the damping of inclination is almost the same as 45 , depending on e0. Still, no inclination increase is observed ◦ for the 1 MJup planet, but with a maximum at i0 ≈ 20 .However, in the high eccentricity simulations (e0 = 0.4). We also want

A124, page 6 of 13 B. Bitsch et al.: Highly inclined and eccentric massive planets. I.

40.08 to stress here that the damping rate significantly increases with 5 MJup, e0=0, i0=40 10 M , e =0, i =40 increasing planetary eccentricity for all planetary masses. 40.06 Jup 0 0 5 MJup damping fit The increase of inclination for high-mass planets due to in- 10 MJup damping fit teractions with the disc has been studied in Lubow & Ogilvie 40.04 (2001). They state that the 1:3 mean-motion resonance also 40.02 acts to increase inclination. This resonance is at rr = 2.08rP, ◦ 40 which clearly is not inside an open gap in the case of i0 = 75 (see Fig. 1). However, the resonances closer to the planet (1:2 39.98 and 2:3) are also not completely inside the gap, so that there Inc [deg] should be some damping effects, but the damping of inclination 39.96 through these resonances is weaker than the increase from the 39.94 1:3 resonance because in total the inclination increases for high inclined planets (Fig. 5). Lubow & Ogilvie (2001)alsoused 39.92 small i for their calculations in order to apply linear theory, 0 39.9 which does not apply for large inclinations. The situation for our 0 10 20 30 40 50 60 70 80 high inclination planets might therefore be completely different time [orbits] from their calculations. 75.2 5 MJup, e0=0, i0=75 75.18 10 MJup, e0=0, i0=75 3.4. Moving planets in discs 5 MJup damping fit 75.16 10 MJup damping fit 3.4.1. Short-term evolution 75.14 In order to verify the results of inclination and eccentricity 75.12 change, we present in this section simulations of planets evolv- 75.1 ing freely in the disc. The planets are moving because of the 75.08 influences of the discs forces. We present here several interest- Inc [deg] ing cases for planets with high inclinations. The first case is for 75.06 ◦ ◦ 5 MJup and 10 MJup with an inclination of i0 = 40 and i0 = 75 75.04 in circular orbits with an evolution time of 80 planetary orbits. 75.02 In Fig. 6 the evolution of inclination with time is presented for 75 the two different planets and inclinations. The evolution is nearly 74.98 identical, as was predicted by the measured forces for the plan- 0 10 20 30 40 50 60 70 80 ets on fixed orbits, which is shown by the solid lines (rates from time [orbits] Fig. 5). One should be aware, however, that by keeping the planet Fig. 6. Evolution of inclination of 5 MJup and 10 MJup planets with an = ◦ = ◦ in a fixed orbit, angular momentum in the system is not con- initial inclination of i0 40.0 (top)andi0 75.0 (bottom) in circular served because, for example, the inclination of the disc is ris- orbits. The simulations have been restarted with a moving planet after the disc was evolved for a fixed planet for 400 planetary orbits. The time ing (see Fig. 3) while the planet remains in a fixed orbit. The ff index has been reset to zero and the lines correspond to the expected e ect of conserving angular momentum is not a problem for damping rates from Fig. 5. low-mass planets, where the measured forces match perfectly with the inclination damping rates for moving planets (Bitsch & Kley 2011), but for big planets of several Jupiter masses this can soon as the eccentricity has dropped to e ≈ 0.4, the inclination lead to small differences because the angular momentum transfer starts to decrease again. from disc to planet and vice-versa is much larger. This exchange of inclination and eccentricity is represen- tative of the Kozai mechanism, introduced initially to describe 3.4.2. Long-term evolution the evolution of a highly inclined asteroid perturbed by Jupiter (Lidov 1962; Kozai 1962). A similar Kozai mechanism affects The long-term evolution of planets with different inclinations the orbits of highly inclined planets with respect to a disc and eccentricities is displayed in Fig. 7. At the beginning of (Terquem & Ajmia 2010; Teyssandier et al. 2013). For inclina- the evolution, the change of inclination and eccentricity matches tions above a critical value, the gravitational force exerted by the those presented in Figs. 4 and 5 for planets in fixed orbits. disc on the planet produces Kozai cycles where the eccentricity However, the evolution after the initial orbits is quite different of the planet can be pumped to high values, in antiphase with its from what was expected by the previous simulations. inclination. We note that the Kozai mechanism is visible in the ◦ In the 10 MJup, e0 = 0.0, i0 = 75 case, the inclination ini- given computation time because of the high mass values consid- tially increases slightly with a rate that corresponds to the pre- ered in our study (5 and 10 MJup), comparable to the total mass of dicted rate (see also Fig. 6). At the same time, the eccentric- the disc (0.01M ). Indeed high masses induce faster dynamical ity of the planet increases and after about 250 orbits it reaches evolution. e ≈ 0.25. This eccentricity corresponds to inclination damping When the planet starts at a larger initial eccentricity (e0 = 0.2 (Fig. 5), which is what happens in the evolution of the planet: the or e0 = 0.4), the general behaviour is similar as can be seen in inclination drops. However, the eccentricity still increases at the Fig. 7, but the first Kozai cycle occurs earlier. Circular orbits at same time, which was not predicted by the analysis of planets in high inclination constitute an unstable equilibrium of the secu- fixed orbits (Fig. 4). The eccentricity then rises until a peak of lar dynamics, so the evolution at zero initial eccentricity remains e ≈ 0.9, where it starts to drop again. At the same time, the incli- for a while close to the separatrix associated with the equilib- nation decrease stops and the inclination starts to rise again. As rium (Libert & Henrard 2007). The Kozai effect does not act for

A124, page 7 of 13 A&A 555, A124 (2013)

80 a full Kozai cycle, probably because their mass-ratio between planet and disc is smaller than in our case. This shows that for 70 ◦ i0 > 40 , the measure of the forces on a planet on a fixed orbit is not relevant. In this case, damped Kozai oscillations will 60 govern the long-term evolution of the orbital parameters. This 50 phenomenon can be of crucial importance for the study of the fate of planets scattered on high-inclination orbits. Inc [deg] 40 3.5. Fitting for e and i 30 In Figs. 5 and 4 we provided the change of di/dt and de/dt for 20 ff 0 100 200 300 400 500 600 700 di erent planetary masses. In these plots, lines indicate a fit for time [orbits] these data points. We now present the fitting formulae, which depend on the planet mass MP, the eccentricity eP, and inclina- 10 MJup, e0=0, i0=75 10 MJup, e0=0, i0=40 10 MJup, e0=0.2, i0=75 5 MJup, e0=0, i0=75 tion iP. The inclination iP used in the presented formulae is given 10 MJup, e0=0.4, i0=75 in degrees, as is the resulting di/dt. As discussed in the previous ◦ 0.9 section, these formulae are only relevant for i0 < 40 where no complex cycles are observed. Therefore, in fitting our parame- 0.8 ters we have ignored the data points corresponding to high incli- 0.7 nations, in particular the ones showing inclination increase. This 0.6 applies to the fitting of inclination and eccentricity. 0.5 As can be seen in the figures, the results of the numerical 0.4 simulations are all but smooth. Therefore, one should not expect

eccentricity the fit to be very accurate with simple functions. However, our 0.3 goal is to catch the big picture, and to provide an acceptable 0.2 order of magnitude of the effect of the disc on the inclination 0.1 and eccentricity. In log scale, the data appear relatively close 0 to an increasing power law of iP for small iP, and a decreasing 0 100 200 300 400 500 600 700 power law of iP for large iP. Therefore, we base our fits on the time [orbits] general form for the damping rates 10 M , e =0, i =75 10 M , e =0, i =40 Jup 0 0 Jup 0 0 M −1/2 10 MJup, e0=0.2, i0=75 5 MJup, e0=0, i0=75 F i = − disc ai−2b + ci−2d , 10 MJup, e0=0.4, i0=75 ( P) P P (11) 0.01 M Fig. 7. Long-term evolution of planets with different inclinations, ec- where b is positive and d is√ negative. This way, for small iP, centricities, and masses in discs. The simulations are started from the F (i ) ≈ ib M /0.01 M a , and for large i , F (i ) ≈ equilibrium structures with fixed planets, where the planets are then re- P P disc√ P P top plot d ffi leased and allowed to move freely inside the disc. The features iP Mdisc/0.01 M c .Thecoe cients a, b, c,andd depend on the inclination of the planets, while the bottom plot shows the eccen- the planet mass and eccentricity, and are fitted to the data as fol- tricity of the planets. In the beginning the changes of eccentricity and lows. The damping rate also has to be linear dependent on the inclination match the ones displayed in Figs. 4 and 5. disc mass Mdisc/M, as our simulations linearly scale with the gas density. initial inclinations smaller than a critical value (i < 40◦ in the 0 3.5.1. Eccentricity restricted problem of Kozai 1962). We therefore also display a = ◦ planet with i0 40 , and show that the eccentricity and inclina- We do not want (de/dt)totendtozerowheniP tends to zero. tion oscillations are significantly reduced. A pure increasing power law of iP is inappropriate here. The Even if eccentricity can be pumped to high values, the Kozai damping function will be given by mechanism only postpones the alignment with the disc and the − F = − Mdisc + −2b + −2d 1/2 circularization of the orbit induced by damping forces of the disc e(iP) a(iP iD) ciP , (12) on the planet (given in Figs. 4 and 5). As clearly shown by the 0.01 M evolution of the planet with e0 = 0.2, Kozai cycles repeat with where iD is a small inclination so that for iP ≈ 0, de/dt ≈ ib reduced intensity. The drop of inclination is much larger than Mdisc √D − .WeareusingiD = M˜ p/3 degrees in Eq. (12), where the raise of inclination, after the eccentricity increase/decrease 0.01 M a ˜ = cycle. These results are in agreement with Teyssandier et al. Mp 1000 Mp/M is the planet mass in Jupiter masses. For = (2013), showing that low-mass planets would remain on in- small eccentricities, it is expected that eP/(de/dt) τe is con- ffi −2 clined and eccentric orbits over the disc lifetime, while higher stant. This makes the coe cient a proportional to eP .Wefind mass planets would align and circularize. We also illustrate in that de/dt is well fitted by the above general form using the coefficients Fig. 7 the influence of the planet mass by considering a planet of − ˜ 5 M (e = 0.0): the eccentricity value reached during the sec- = 2 − 2 ˜ Mp + ˜ − ˜ 2 Jup ae(MP, eP) 80 e exp e Mp/0.26 15 20 11Mp Mp ond cycle of inclination increase (at 450 orbits) is higher for the P P = ˜ 5 MJup planet, as expected. be(MP) 0.3Mp ff M˜ p The e ect of Kozai oscillations between a disc and planet ce(MP) = 450 + 2 was also stated in Xiang-Gruess & Papaloizou (2013)however, Xiang-Gruess & Papaloizou (2013) were not able to resolve de(MP) = −1.4 + M˜ p/6. (13)

A124, page 8 of 13 B. Bitsch et al.: Highly inclined and eccentric massive planets. I.

40 0 The second degree polynomial function of M˜ p in the expression of a is just a reﬁnement, its value being between 30 and 50 for 35 -0.005 -0.01 1 < M˜p < 10. We note, however, that it is negative for M˜p > 12 30 M˜ < -0.015 so this expression only applies for p 11, but this covers the 25 range of giant planets. To describe the change of eccentricity -0.02 20 -0.025 we add a second function Ge, which describes the eccentricity i in deg -0.03 increase for high-mass planets. The damping and excitation of eP 15

ff -0.035 di/dt in [deg/orbit] are two di erent mechanisms that add on the planet, and one 10 of them finally dominates, setting the sign of de/dt.Thefitsin -0.04 Fig. 4 are the added functions. 5 -0.045 For Ge we use the result of Papaloizou et al. (2001) who cal- 0 -0.05 ◦ 0 0.1 0.2 0.3 0.4 0.5 culated the eccentricity excitation for i0 = 0 high mass-planets. e Our calculation is presented in Appendix C and gives 40 0 -0.005 MP Mdisc 35 G | = = . e . e i 0 12 65 2 P (14) -0.01 M 30 -0.015 Then, we find that this excitation decreases with i as a Gaussian 25 -0.02 function, finally making 20 -0.025

⎛ ⎞ i in deg -0.03 M M ⎜ (i /1◦) 2⎟ 15

⎜ ⎟ di/dt in [deg/orbit] G (i , M , e ) = 12.65 P disc e exp ⎝⎜− P ⎠⎟ · (15) -0.035 e P P P 2 P ˜ 10 M Mp -0.04 5 -0.045 In principle planets with M < 5 M and e < 0.3donot P Jup 0 -0.05 show any signs of eccentricity increase and the Gaussian func- 0 0.1 0.2 0.3 0.4 0.5 tion should not be added in that case. However, the function is e designed to scale with the planetary mass, so that lower mass ff ff Fig. 8. Top:di/dt for a 5 MJup planet with di erent eccentricities and planets are not a ected by it. The change of eccentricity is then inclinations. The values of di/dt have been determined with the formula given by the sum of F and G . e e given in Sect. 3.5. Bottom: same, but for 10 MJup. We note the different colour coding as the change is dependent on the planetary mass. 3.5.2. Inclination In the case of inclination damping data, we notice that the decreasing power law dominates actually before the intersection Figure 8 clearly indicates that the damping rate of inclination −2b is highest for planets with a large eccentricity that are moderately with the increasing power law; thus, we multiply the term aiP ◦ inclined above the midplane (iP ≈ 15 ). The inclination damping in our general formula by a Gaussian function of iP centred on 0◦, so that this term is not affected for small i but van- rate indicates that planets that are scattered during the gas disc P i < ◦ ishes more quickly than is natural. It allows our fitting formula phase in orbits with moderate inclination ( P 40 ), would lose to catch the peak of damping in inclination observed around 5 to their inclination well within the gas dispersal of the disc. 20 degrees in Fig. 5.Forsmalli ,di/dt should be close to lin- As already indicated in Fig. 4, the eccentricity is always P damped for high inclinations. For high planetary masses, the ear in iP,sothecoefficient b should be close to 1. The damping function for inclination F is then given, in degrees per orbit, by eccentricity of the planet can increase for low planetary incli- i nations because of interactions with disc. We find an eccen- = × 4 − ˜ 3 ai(MP, eP) 1.5 10 (2 3eP)Mp tricity increase for both high-mass cases, but the increase of = + ˜ 2 eccentricity declines with increasing eccentricity and inclina- bi(MP, eP) 1 MpeP/10 tion. Additionally, the threshold of eP and iP for which eccen- 6 2 3 tricity can increase is larger for higher mass planets, which is ci(MP, eP) = 1.2 × 10 / (2 − 3eP) 5 + e M˜ p + 2 P indicated by the white line in Fig. 9 that represents the transition = − + di(eP) 3 2eP from eccentricity increase to decrease. Below the line eccentric- ◦ ity increases, above the line eccentricity decreases. gi(MP, eP) = 3M˜ p/(eP + 0.001) × 1 −2bi F = − Mdisc iP − 2 i(MP, eP, iP) ai ◦ exp( (iP/gi) /2) (16) 4. Application to single-planet systems 0.01 M 1 − −1/2 i 2di The movement of a single planet in the disc can only be pre- + c P · ◦ i 40◦ dicted if i < 40 and e < 0.65 as the planet would undergo a Kozai-oscillation for larger i. Additionally, the fitting formula We note that the expression for coefficient ci is clearly not valid might not be totally accurate for e > 0.5, since our simulations for e > 2/3. only cover an eccentricity space of up to e = 0.4. In Fig. 10 the ◦ From our formulae for de/dt and di/dt we can now estimate trajectory of the 10 MJup planet with i0 = 75 and e0 = 0.4, how the eccentricity and inclination of a planet will evolve for whichwasshowninFig.7 is displayed. This illustrates that all eP and iP.InFig.8 the di/dt for different inclinations and the movement of the planet is a complex process as long as the ◦ eccentricities for 5 MJup and 10 MJup according to the formulae is Kozai-oscillations are still operational, but as soon as i < 40 ,the presented. In Fig. 9 the de/dt for the same two planetary masses planet loses inclination, which is then not converted back into ec- is plotted. centricity. The planet is damped towards midplane on a non-zero

A124, page 9 of 13 A&A 555, A124 (2013)

40 0.001 80 0.001 35 0 70 0 30 -0.001 60 -0.001 25 -0.002 50 20 -0.003 40 -0.002 de/dt de/dt i in deg 15 -0.004 i in deg 30 -0.003 10 -0.005 20 -0.004 5 -0.006 10 0 -0.007 0 -0.005 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 e e 40 0.001 Fig. 10. Evolution of e and i of the 10 M planet (shown in Fig. 7) 35 0 Jup with i = 75◦ and e = 0.4inthee-i plane (black line). The background -0.001 0 0 30 is the extended formula of the ﬁt for de/dt and the white line marks -0.002 25 the transition between eccentricity increase and damping as in Fig. 8. -0.003 20 The blue lines indicate calculated trajectories of 10 MJup planets from

-0.004 de/dt

i in deg the fitting formulae. 15 -0.005 10 -0.006 5 -0.007 the planet have been calculated. By using these forces, a change 0 -0.008 of de/dt and di/dt has been determined. 0 0.1 0.2 0.3 0.4 0.5 0.6 Inclination and eccentricity are in general damped by the in- e teractions with the disc. For 1 MJup the damping rate of e and i ff ◦ Fig. 9. Top:de/dt for a 5 MJup planet with di erent eccentricities and in- is highest for only very small inclinations (i0 ≈ 3 ), while the clinations. The values of de/dt have been determined with the formula maximal damping rate is shifted to larger inclinations for more giveninSect.3.5. Bottom: same, but for 10 MJup. The white line in the massive planets. As the more massive planets carve deeper gaps figure indicates the transition between eccentricity increase and eccen- inside the disc, the damping interactions with the disc are re- tricity damping. Below the white line, the eccentricity increases, above the line eccentricity decreases. We note the different colour coding as duced. But for larger inclinations, the planet can feel the full the change is dependent on the planetary mass. damping potential of the disc and is therefore damped in e and i at a faster rate. There are two exceptions. The first is for low-inclination ffi − eccentricity. This non-zero eccentricity will actually hold in this planets with a su cient mass (MP > 4 5 MJup). In this case, case (see Sect. 3.3.1). the interactions of the planet with the disc result in an increase of eccentricity of the planet, which has already been observed A typical damping rate of de/dt = 0.001/orbit would sug- and studied (Papaloizou et al. 2001; Kley & Dirksen 2006). gest that the planet will lose ≈0.085 in eccentricity in the period However, our 3D results predict an increase of eccentricity for of 104 years. A typical damping rate of di/dt = 0.01 deg/orbit in- ◦ lower planetary masses than the previous studies. dicates that the planet will lose ≈8.5 of inclination in 104 years. The second exception arises for massive planets (M ≈ The important parameters for the evolution of the orbit P Mdisc, in our case for MP > 5 MJup) on high initial inclinations of a planet are the damping timescales τe = e/(de/dt)and ◦ (i0 > 40 ). In the long-term evolution of the planet, eccentricity τi = i/(di/dt). We find that e/Fe is much smaller than i/Fi ◦ can increase, while inclination is damped and vice-versa. The for i > 10−20 , depending on the planet mass and eccentric- planet undergoes a Kozai-cycle with the disc, but in time the os- ity. Thus, planets scattered on highly inclined orbits will follow cillations of the planet in e and i diminish, as e and i get damped a certain pattern. While the inclination is damped slowly and still by the disc at the same time. The planet will end up in midplane high, the eccentricity is damped to zero. After the inclination is through the interactions with the disc. damped further, the eccentricity of the planet can rise because of interactions with the disc (if e is below the white line in Fig. 10). In Sect. 3.5 we provided formulae for di/dt and de/dt for Finally the inclination is damped to zero and the planet remains high-mass planets, which we fitted to the numerical hydrody- with a non-zero eccentricity. This is illustrated by the blue lines namical simulations. The formulae can now be used to calculate the long-term evolution of planetary systems during the in Fig. 10 that correspond to calculated trajectories of 10 MJup planets. gas phase of the disc with N-Body codes. However, we recom- mend not using the fitting formula, if the planetary eccentricity Nevertheless, this suggests that at the time of the disc disper- is e > 0.65 and if i > 40◦ (because of the Kozai interactions, a sal, the favoured endstate for the planet’s evolution is an eccentric orbit in midplane of the disc. This implies that the scattering fit that can be used for the long-term evolution of planets is hard to predict). process of inclined planets must have taken place after the gas is depleted or gone. In the end, the planet’s inclination will be damped to zero. Low-mass planets (MP < 4−5 MJup) will end up in circular orbits in the midplane of the disc, while higher mass planets (MP > 5. Summary 5 MJup) will pump their eccentricity to larger values because of interactions with the disc. This implies that the scattering process We have presented the evolution of eccentricity e and inclina- of inclined planets must have taken place after the gas is well tion i of high-mass planets (MP ≥ 1 MJup)inisothermalpro- depleted. toplanetary discs. The planets have been kept on fixed orbits The influence of the gaseous protoplanetary disc on the in- around the host star, and the forces from the disc acting onto clination is also of crucial importance, if multiple planets are

A124, page 10 of 13 B. Bitsch et al.: Highly inclined and eccentric massive planets. I. present in the disc that excite each other’s inclination during their 1e-07 migration (Libert & Tsiganis 2009, 2011a). The inﬂuence of the 9e-08 disc on the long-term evolution of multi-body systems will be 8e-08 studied in a future paper. 7e-08

Acknowledgements. B. Bitsch has been sponsored through the Helmholtz 6e-08

Alliance Planetary Evolution and Life. The work of A.-S. Libert is supported by N 5e-08 an FNRS Postdoctoral Research Fellowship. The calculations were performed F on systems of the Computer centre (ZDV) of the University of Tübingen and 4e-08 systems operated by the ZDV on behalf of bwGRiD, the grid of the Baden Württemberg state. We thank the referee Willy Kley for his useful and helpful 3e-08 remarks that improve the paper. 2e-08

1e-08 Δ t = 1.670E-3 orbits Δ t = 5.598E-4 orbits 0 Appendix A: Additional information on numerics 0 10 20 30 40 50 60 70 In principle, a fast vertical movement (more than one grid cell time [orbits] ◦ per timestep) through the grid could cause problems with the Fig. A.1. FN for 5 MJup planets with i = 75 on circular orbits. The Fargo algorithm, as Fargo shifts the grid cells for several cells planet’s feature different time steps, as indicated in the plot. FN has azimuthally and the effects of the planet on the gas might get been averaged over one running orbit. corrupted. In Fig. A.1 we present the evolution of the normal component of the disturbing force FN (which has been averaged over one orbit) of planets with i = 75◦ on circular orbits. The 0.003 ff 5 MJup, e=0.0, rsm=0.8 two simulations shown feature di erent time-step lengths. For 5 M , e=0.0, r =0.5 Jup ε sm the simulation with larger timestep, the planet moves through 0.002 10 MJup, e=0.0, -pot, rsm=0.8 about one vertical grid cell in each time step. For the shorter 10 MJup, e=0.0, rsm=0.8 timestep, three timesteps are needed to cover the vertical extent 0.001 of one grid cell. The evolution of FN seems to be identical, indicating that the length of the timestep is not of crucial importance 0 here also because we use a rotating frame so that the planet is on a fixed position inside the numerical grid where the Fargo -0.001 algorithm does not shift grid cells for r ≈ aP. di/dt [deg/orbits] The smoothing of the planetary potential is crucial for avoid- -0.002 ing singularities at the planet’s location. In Sect. 2.1 the numerical potential for the planets was introduced. Of crucial impor- -0.003 tance here is the smoothing length rsm. A smaller smoothing length rsm leads to a deeper planetary potential. This in turn leads -0.004 to a larger accumulation of mass at the planet’s location, but this 0 10 20 30 40 50 60 70 80 increase in density near the planet can be very high for large inclination in degrees planets, especially in the isothermal case. This increase of den- Fig. A.2. Change of inclination di/dt for 5 MJup and 10 MJup planets in sity near the planet is unphysical, as normally the temperature circular orbits for two different smoothing length for the planetary po- and pressure gradients should stop the accumulation of gas at tential. The cubic potential is used for all displayed simulations, unless some point, which is not possible in the isothermal case. In this stated otherwise. situation, the density can become so large that the gradients of density near the planet become too steep and the timestep inside the code collapses down to very small values, which makes an 0 integration over several orbits impossible. We therefore return to -1e-07 the -potential for the 10 M planet. Jup -2e-07 In Fig. A.2 we present the inclination damping for 5 MJup planets in circular orbits for two different smoothing lengths, -3e-07 rsm = 0.8andrsm = 0.5. Changing the planetary potential seems -4e-07 ±

to influence the damping of inclination by up to 15%, but the N -5e-07 general trend is the same. Even with a deeper planetary poten- F tial, the inclination of a planet seems to increase for large initial -6e-07 inclinations. The main difference seems to be that no inclination -7e-07 ◦ increase can be observed for the i0 = 55 case with a smoothing -8e-07 length of rsm = 0.5. This has also been observed for the 10 MJup planet where the difference between the depth of the two po- -9e-07 260 x 32 x 384 390 x 48 x 576 tentials is supposed to be stronger (as we change from the to -1e-06 the cubic potential), but the trend is the same as for the 5 MJup 0 20 40 60 80 100 ◦ planet. For i0 = 75 the inclination seems to increase for models time [orbits] of planets in fixed orbits for both 5 M and 10 M .Wethere- Jup Jup = ◦ = fore conclude that the general trend is conserved regardless of Fig. A.3. FN for a 10 MJup planet with i 3 in a e0 0.4 orbit. FN has the chosen planetary potential and smoothing length. been averaged over one running orbit.

A124, page 11 of 13 A&A 555, A124 (2013)

In order to find the sufficient numerical resolution for our where simulations of inclination damping, we have performed several u = sin θ cos φu + sin θ sin φu + cos θu resolution tests. In Fig. A.3 we present the results of these tests. r x y z uθ = cos θ cos φux + cos θ sin φuy − sin θuz The plotted quantity FN has been averaged over one running or- = − + bit. Keep in mind that FN has been averaged over 40 orbits to uφ sin φux cos θuy, (B.7) determine the change of inclination in the end. The simulations with the angles θ and φ of the grid cell, which differ for each grid = ◦ feature a 10 MJup planet with i 3 , so it is well embedded inside cell. This gives us for L in Cartesian coordinates the disc. The numerical resolution of the grid has been changed L = (cos θ cos φL − sin φL )u from 260×32×384 to 390×48×576. As the crucial force FN for θ φ x inclination damping gives the same results for both resolutions, +(cos θ sin φLθ + cos φLφ)uy − sin θLθuz. (B.8) we use the lower resolution for our simulation with confidence. For the inclination of the disc we now take a mass-averaged specific angular momentum (averaged in polar and azimuthal direction) Appendix B: Eccentricity and inclination of the disc ΣLspec.,cmc L (r) = , (B.9) To determine the eccentricity and inclination of the disc, we take av. Σm a mass-weighted average of the eccentricity of all grid cells. To c compute the eccentricity we take the specific total energy (in where the subscript c denotes the grid cell number, and mc the mass units) corresponding mass of the grid cell. Now we can compute the angle between Lav. and the z-axis, which gives us the averaged GM 1 inclination at each ring of the disc. E = − + u2, (B.1) tot,spec. r 2 Appendix C: Increase of eccentricity where u is the velocity vector of a given grid cell and r = x2 + y2 + z2 the radial component towards the grid cell. The We follow Papaloizou et al. (2001) to calculate the maximum total energy is given by value of eccentricity increase for high-mass planets, as it is given by Ag in Eq. (15). In Papaloizou et al. (2001) the increase of GM eccentricity is calculated through the growth rates of the modes E = − , (B.2) tot 2a of the Lindblad resonance, which is given by 1 dJ where a is the semi-major axis towards the grid cell. With that, γ = , (C.1) 4J dt we can compute a where GM 1 GM − + u2 = − J = −1 2 − 1 Σ 2 3Ω MPeP (GM)rP edr drdφ, (C.2) r 2 2a 2 2 GM 1 GM where e is the disc’s eccentricity and r the planetary distance ⇒ a = − u2 − · (B.3) d P 2 2 r to star. The integral basically gives the disc mass, which is comparable to the planet’s mass, but as ed is much smaller than eP With a we can now compute the eccentricity e of each grid cell: (see Fig. 3), the term concerning the disc eccentricity is much smaller than the term concerning the planetary eccentricity. We 2 therefore choose to neglect it in our estimate of the eccentricity Lspec. L = GM a(1 − e2) ⇒ e = 1 − , (B.4) increase. We then get spec. GM a dJ = −MPe˙PeP (GM)rP, (C.3) where Lspec. = r × u is the specific angular momentum of each dt grid cell. To get an estimate of the eccentricity of the disc, we which leads to √ make a mass-weighted average of the eccentricity of each grid −M e˙ e (GM )r 1˙e γ = P P P √ P = P · (C.4) cell (averaged in azimuthal and polar coordinates) in order to 1 2 −4( MPe (GM)rP) 2eP get edisc(r): 2 P As also Σmθφeθφ 8 = γ Mdisc MP rP edisc(r) Σ , (B.5) = mθφ ω M2 r 2 where m is the mass of the grid cell. − 2r d¯ed r − 21 + 5 2 θφ 9π (re¯d 3 dr ) r 4 eP 1 7 (rP/r) Because we use spherical coordinates r, θ, φ for the inclina- × P , (C.5) e2 + 2πΣe2r3Ωdr/ M ωr2 tion, we have to transform Lspec. first into Cartesian coordinates P d P P in order to calculate the mass average. This has to be done be- where we set ed = 0.0and¯ed = 0.0 because we are only inter- cause each product L = r × u is given in a different local coordi- ested in a first order estimate of the eccentricity increase from nate system of each grid cell, but for the average the angular mo- = 3 = mentum vectors should always be in the same coordinate frame. the disc. With ω GM/rP we find fore ˙P 2ePγ The angular momentum vector is given in the two coordinate = 2 = G | systems by e˙P 12.65MPMdisc/M eP e i=0, (C.6) which gives the increase of eccentricity for a planet orbiting in = + + L Lrur Lθuθ Lφuφ the midplane of the disc, which fits quite well with the results of L = Lxux + Lyuy + Lzuz, (B.6) our simulations (Fig. 4).

A124, page 12 of 13 B. Bitsch et al.: Highly inclined and eccentric massive planets. I.

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